Benchmark problems for phase retrieval

In recent years, the mathematical and algorithmic aspects of the phase retrieval problem have received considerable attention. Many papers in this area mention crystallography as a principal application. In crystallography, the signal to be recovered is periodic and comprised of atomic distributions arranged homogeneously in the unit cell of the crystal. The crystallographic problem is both the leading application and one of the hardest forms of phase retrieval. We have constructed a graded set of benchmark problems for evaluating algorithms that perform this type of phase retrieval. The data, publicly available online, is provided in an easily interpretable format. We also propose a simple and unambiguous success/failure criterion based on the actual needs in crystallography. Baseline runtimes were obtained with an iterative algorithm that is similar but more transparent than those used in crystallography. Empirically, the runtimes grow exponentially with respect to a new hardness parameter: the sparsity of the signal autocorrelation. We also review the algorithms used by the leading software packages. This set of benchmark problems, we hope, will encourage the development of new algorithms for the phase retrieval problem in general, and crystallography in particular.

💡 Research Summary

The paper addresses the phase‑retrieval problem as it appears in X‑ray crystallography, where the unknown signal is a periodic electron‑density function defined on a crystal lattice. While many recent works on phase retrieval focus on non‑periodic or sparsity‑constrained signals, the authors argue that crystallographic phase retrieval presents unique challenges due to the inherent periodicity and the structure of the autocorrelation function.

First, the authors review the mathematical formulation of crystallographic phase retrieval. The measured data are the magnitudes of the Fourier transform of the electron density, |ρ̂(q)|², sampled on the reciprocal lattice Λ*. The autocorrelation a(y)=∫ρ(x)ρ(x+y)dx is the inverse Fourier transform of these magnitudes and contains all pairwise inter‑atomic distances. For a periodic crystal, the autocorrelation peaks are densely packed within a single unit cell, leading to a number of peaks proportional to N² (where N is the number of atoms) rather than N for an aperiodic molecule. Consequently, the sparsity of the autocorrelation, measured as N²/M (M being the number of resolution elements), is a far more accurate predictor of problem hardness than the usual signal‑sparsity N/M.

To enable systematic evaluation of algorithms, the authors construct a graded benchmark suite. They generate synthetic crystals with varying numbers of atoms, unit‑cell sizes, and resolution levels, thereby controlling the autocorrelation sparsity. The benchmark set is divided into five difficulty levels; each level contains dozens of instances with publicly available files (atom coordinates, Fourier magnitudes, autocorrelation data). A success criterion is defined that mirrors crystallographic practice: recovered atom positions must be accurate to within typical experimental tolerances (≈0.5 Å) and the reconstructed electron density must be visually consistent with the ground truth.

The paper then surveys the algorithms used in the leading commercial packages (SHELX, PHENIX, CNS, etc.). These tools rely on heuristics developed in the 1970s—alternating projections, direct methods, and refinement cycles—that are highly engineered but difficult to compare fairly because of proprietary implementations and numerous tunable parameters.

As a transparent baseline, the authors implement a simple iterative scheme called Reflect‑Reflect‑Reflect (RRR). RRR alternates projections onto the Fourier‑magnitude constraint set and onto a sparsity‑promoting real‑space constraint, using a fixed relaxation parameter. Despite its simplicity, RRR reproduces the performance trends of more sophisticated software on the benchmark instances. Empirically, the runtime of RRR grows exponentially with the autocorrelation‑sparsity index. This observation supports the authors’ claim that the hardness of crystallographic phase retrieval scales with N²/M rather than N/M.

The authors also discuss the relationship between crystallographic phase retrieval and the “bit‑retrieval” problem from theoretical computer science. In the aperiodic case, the autocorrelation polynomial a(x)=b(x)b(1/x) can be factored efficiently using lattice‑basis reduction (LLL), but for periodic signals the relevant polynomial ring Z